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José Carlos de Albuquerque

Publications and source records attributed to José Carlos de Albuquerque.

10 recordsLinked to original sources
arXiv2026

Minimax solutions for FitzHugh-Nagumo systems with critical exponential growth

We study a class of FitzHugh-Nagumo type systems with critical exponential growth in a smooth bounded domain of $\mathbb{R}^{2}$. The classical reduction method turns the strongly indefinite system into a scalar nonlocal problem whose quadratic part induces a norm equivalent to, but different from the standard Sobolev norm, so that the Trudinger-Moser threshold must be handled in this framework. We prove the existence of a ground state solution via a mountain pass argument combined with a Nehari level comparison. We also prove the existence, without imposing the monotonicity condition required in the ground state argument, of a nontrivial solution obtained through a saddle-point linking argument based on the spectral decomposition of $\operatorname{T}=-Δ+B$.

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arXiv2021

Fractional magnetic Schrödinger equations with potential vanishing at infinity and supercritical exponents

This paper focuses on the following class of fractional magnetic Schrödinger equations \begin{equation*} (-Δ)_{A}^{s}u+V(x)u=g(\vert u\vert^{2})u+λ\vert u\vert^{q-2}u, \quad \mbox{in } \mathbb{R}^{N}, \end{equation*} where $(-Δ)_{A}^{s}$ is the fractional magnetic Laplacian, $A :\mathbb{R}^N \rightarrow \mathbb{R}^N$ is the magnetic potential, $s\in (0,1)$, $N>2s$, $λ\geq0$ is a parameter, $V:\mathbb{R}^N \rightarrow \mathbb{R}$ is a potential function that may decay to zero at infinity and $g: \mathbb{R}_{+} \rightarrow \mathbb{R}$ is a continuous function with subcritical growth. We deal with supercritical case $q\geq 2^*_s:=2N/(N-2s)$. Our approach is based on variational methods combined with penalization technique and $L^{\infty}$-estimates.

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arXiv2021

Existence of solutions for a fractional Choquard--type equation in $\mathbb{R}$ with critical exponential growth

In this paper we study the following class of fractional Choquard--type equations \[ (-Δ)^{1/2}u + u=\Big( I_μ\ast F(u)\Big)f(u), \quad x\in\mathbb{R}, \] where $(-Δ)^{1/2}$ denotes the $1/2$--Laplacian operator, $I_μ$ is the Riesz potential with $0<μ<1$ and $F$ is the primitive function of $f$. We use Variational Methods and minimax estimates to study the existence of solutions when $f$ has critical exponential growth in the sense of Trudinger--Moser inequality.

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arXiv2020

Compact embedding theorems and a Lions' type Lemma for fractional Orlicz-Sobolev spaces

In this paper we are concerned with some abstract results regarding to fractional Orlicz-Sobolev spaces. Precisely, we ensure the compactness embedding for the weighted fractional Orlicz-Sobolev space into the Orlicz spaces, provided the weight is unbounded. We also obtain a version of Lions' "vanishing" Lemma for fractional Orlicz-Sobolev spaces, by introducing new techniques to overcome the lack of a suitable interpolation law. Finally, as a product of the abstract results, we use a minimization method over the Nehari manifold to prove the existence of ground state solutions for a class of nonlinear Schrödinger equations, taking into account unbounded or bounded potentials.

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arXiv2019

On the extreme value of the Nehari manifold method for a class of Schrödinger equations with indefinite weight functions

In this work we are concerned with the following class of equations \[ -Δ_p u -λh(x)|u|^{p-2}u=f(x)|u|^{γ-2}u, \quad \mbox{in } \mathbb{R}^N, \] involving indefinite weight functions. The existence of solution may depend on the parameter $λ$. We analyze the extreme value $λ^{*}$ and study its relation with the Nehari manifold. Our goal is to establish the existence of two solutions when $λ>λ^{*}$. This work extends and complements the results obtained by J. Chabrowski and D.G. Costa [Comm. Partial Differential Equations 33 (2008), 1368--1394]

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arXiv2018

Ground states for a linearly coupled system of Schrödinger equations on $\mathbb{R}^{N}$

We study the following class of linearly coupled Schrödinger elliptic systems $$\left\{ \begin{array}{lr} -Δu+V_{1}(x)u=μ|u|^{p-2}u+λ(x)v, & \quad x\in\mathbb{R}^{N}, \\ -Δv+V_{2}(x)v=|v|^{q-2}v+λ(x)u, & x\in\mathbb{R}^{N}, \end{array} \right. $$ where $N\geq3$, $2 0$ such that the coupled system possesses positive ground state solution for all $μ\geqμ_{0}$. In these cases, we use a minimization method based on Nehari manifold. Finally, we consider the case $p=q=2^{*}$, and we prove that the coupled system has no positive solutions. For that matter, we use a Pohozaev identity type.

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arXiv2018

Existence of bound and ground states for fractional coupled systems in $\mathbb{R}^{N}$

In this work we consider the following class of nonlocal linearly coupled systems involving Schrödinger equations with fractional laplacian $$ \left\{ \begin{array}{lr} (-Δ)^{s_{1}} u+V_{1}(x)u=f_{1}(u)+λ(x)v, & x\in\mathbb{R}^{N}, (-Δ)^{s_{2}} v+V_{2}(x)v=f_{2}(v)+λ(x)u, & x\in\mathbb{R}^{N}, \end{array} \right. $$ where $(-Δ)^{s}$ denotes de fractional Laplacian, $s_{1},s_{2}\in(0,1)$ and $N\geq2$. The coupling function $λ:\mathbb{R}^{N} \rightarrow \mathbb{R}$ is related with the potentials by $|λ(x)|\leq δ\sqrt{V_{1}(x)V_{2}(x)}$, for some $δ\in(0,1)$. We deal with periodic and asymptotically periodic bounded potentials. On the nonlinear terms $f_{1}$ and $f_{2}$, we assume "superlinear" at infinity and at the origin. We use a variational approach to obtain the existence of bound and ground states without assuming the well known Ambrosetti-Rabinowitz condition at infinity. Moreover, we give a description of the ground states when the coupling function goes to zero.

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arXiv2018

Positive ground states for a class of superlinear $(p,q)$-Laplacian coupled systems involving Schrödinger equations

We study the existence of positive solutions for the following class of $(p,q)$-Laplacian coupled systems \[ \left\{ \begin{array}{lr} -Δ_{p} u+a(x)|u|^{p-2}u=f(u)+ αλ(x)|u|^{α-2}u|v|^β, & x\in\mathbb{R}^{N}, -Δ_{q} v+b(x)|v|^{q-2}v=g(v)+ βλ(x)|v|^{β-2}v|u|^α, & x\in\mathbb{R}^{N}, \end{array} \right. \] where $N\geq3$ and $1\leq p\leq q<N$. Here the coefficient $λ(x)$ of the coupling term is related with the potentials by the condition $|λ(x)|\leqδa(x)^{α/p}b(x)^{β/q}$ where $δ\in(0,1)$ and $α/p+β/q=1$. We deal with periodic and asymptotically periodic potentials. The nonlinear terms $f(s), \; g(s)$ are "superlinear" at $0$ and at $\infty$ and are assumed without the well known Ambrosetti-Rabinowitz condition at infinity. Thus, we have established the existence of positive ground states solutions for a large class of nonlinear terms and potentials. Our approach is variational and based on minimization technique over the Nehari manifold.

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arXiv2017

Coupled elliptic systems involving the square root of the Laplacian and Trudinger-Moser critical growth

In this paper we prove the existence of a nonnegative ground state solution to the following class of coupled systems involving Schrödinger equations with square root of the Laplacian $$ \left\{ \begin{array}{lr} (-Δ)^{1/2}u+V_{1}(x)u=f_{1}(u)+λ(x)v, & x\in\mathbb{R}, (-Δ)^{1/2}v+V_{2}(x)v=f_{2}(v)+λ(x)u, & x\in\mathbb{R}, \end{array} \right. $$ where the nonlinearities $f_{1}(s)$ and $f_{2}(s)$ have exponential critical growth of the Trudinger-Moser type, the potentials $V_{1}(x)$ and $V_{2}(x)$ are nonnegative and periodic. Moreover, we assume that there exists $δ\in (0,1)$ such that $λ(x)\leqδ\sqrt{V_{1}(x)V_{2}(x)}$. We are also concerned with the existence of ground states when the potentials are asymptotically periodic. Our approach is variational and based on minimization technique over the Nehari manifold.

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